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March, 1991 Estimating Covariance Matrices
Wei-Liem Loh
Ann. Statist. 19(1): 283-296 (March, 1991). DOI: 10.1214/aos/1176347982


Let $S_1$ and $S_2$ be two independent $p \times p$ Wishart matrices with $S_1 \sim W_p(\sum_1, n_1)$ and $S_2 \sim W_p(\sum_2, n_2)$. We wish to estimate $(\sum_1, \sum_2)$ under the loss function $L(\hat{\sum}_1, \hat{\sum}_2; \sum_1, \sum_2) = \sum_i\{\operatorname{tr}(\sum^{-1}_i \hat{\sum}_i) - \log|\sum^{-1}_i\hat{\sum}_i| - p\}$. Our approach is to first utilize the principle of invariance to narrow the class of estimators under consideration to the equivariant ones. The unbiased estimates of risk of these estimators are then computed and promising estimators are derived from them. A Monte Carlo study is also conducted to evaluate the risk performances of these estimators. The results of this paper extend those of Stein, Haff, Dey and Srinivasan from the one sample problem to the two sample one.


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Wei-Liem Loh. "Estimating Covariance Matrices." Ann. Statist. 19 (1) 283 - 296, March, 1991.


Published: March, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0742.62054
MathSciNet: MR1091851
Digital Object Identifier: 10.1214/aos/1176347982

Primary: 62F10
Secondary: 62C99

Keywords: Covariance matrix , equivariant estimation , Stein's loss , unbiased estimate of risk , Wishart distribution

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 1 • March, 1991
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